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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Einstein solid in equilibrium with a heat bath</h3>


<p class="header_title">Introduction</p>

<p>Consider a simple system, commonly known as an Einstein or harmonic solid. The energy of each particle in an Einstein solid is restricted to the positive integers. That is, each particle may have energy 0, 1, 2, &#8230; The particles do not interact. These particles are equivalent to the
quanta of the harmonic oscillator, which have energy
&#949;<sub>n</sub> = (n + 1/2)h&#957;. If we measure the energies from the
lowest energy state, h&#957;/2, and choose units
such that
h&#957; = 1, we have &#949;<sub>n</sub> = n.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;In the following we will explore the properties of a Einstein solid in equilibrium with a heat bath at temperature T. We will also compare our results with analytical calculations of the thermodynamic properties of the Einstein solid.</p>

<p class="header_title">Algorithm</p>

<p>The program implements the Metropolis algorithm by choosing a particle at random and
randomly increasing or decreasing its energy by one unit. If the energy
is decreased, the change is accepted. If the energy is increased, the program
generates a number
r at random in the unit interval and accepts the change if r
&#8804; e<sup>-&#946;1;</sup>, where &#946; = 1/T. (As usual, we choose units such
that Boltzmann's constant k = 1.) If a trial change is not accepted, the
existing microstate is counted in all averages.</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.einsteinsolid.EinsteinsolidMCApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title">Problems</p>

<ol>

<li>Start from the formal expression for the partition function and find an analytical expression for the mean energy and specific heat of the Einstein solid.</li>

<li>Use the default values for the number of particle N in the solid and the temperature T. Run for a sufficient number of Monte Carlo steps per particle so that the mean energy becomes well defined. When you are satisfied that it is, press the <tt>Accept E and C<sub>V</sub></tt>  button. Then consider a range of values of T and determine the T-dependence of the mean energy and the specific heat.</li>

<li>What is the limiting value of the mean energy and the specific heat for high and low temperatures?</li>

</ol>

<p class="header_title">References</p>

<ul>

<li>Harvey Gould and Jan Tobochnik, <i>Statistical and Thermal Physics,</i> Chapter 4, online notes.</li>

<li>Daniel V. Schroeder, <i>An
Introduction to Thermal Physics,</i> Addison-Wesley (1999).</li>

</ul>

<p class="header_title">Java Classes</p>

<ul>

<li>EinsteinSolid</li>

<li>EinsteinSolidMCApp</li>

</ul>

<p class = "small">Updated 19 March 2007.</p>
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